Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 201 / 6

A real Lévy process satisfies $X_0=0$ almost surely, has stationary independent increments, is stochastically continuous, and is taken in its almost surely càdlàg version.

The Lévy–Khintchine theorem states that there is a unique triplet $(a,b,K)$, where $a\in \mathbb{R}$, $b\ge 0$, and $K$ is a measure on $\mathbb{R}\setminus \{0\}$ satisfyingsuch thatConversely every such triplet is the characteristic triplet of a Lévy process.

Let $B$ be Brownian motion and let $N(ds,dx)$ be an independent Poisson random measure with intensity $dsK(dx)$. Writing $\tilde{N}=N-dsK(dx)$ for its compensated version, the Lévy–Itô decomposition constructsThe four terms are independent drift, Gaussian, compensated small-jump and compound-Poisson large-jump components.

Almost surely differentiable paths must be continuous, so the jump measure must vanish: $K=0$. A nonzero Brownian component has almost surely nowhere-differentiable paths, so also $b=0$. Conversely, if $b=0$ and $K=0$, then $X_t=at$ is differentiable. Thus

The Brownian and drift components are continuous. Any nonzero Lévy measure produces jumps: a set bounded away from zero with positive finite $K$-measure gives a nontrivial compound Poisson component, and increasing such sets detects every nonzero $K$. Hence paths are almost surely continuous exactly when

The compensated small-jump integral is integrable after localization and has mean zero, while the number of large jumps on a compact time interval is finite. Its absolute first moment is finite exactly when the large-jump sizes have finite first moment. Thus $(X_t)$ is integrable exactly when

The Brownian component has finite variance $bt$, and the compensated jump integral has variancewhen this integral is finite. Conversely, a finite second moment forces the jump measure to have a finite second moment. Therefore $(X_t^2)$ is integrable exactly whenThese four equivalences are the Path and moment criteria from a Lévy triplet.